An inspired problem (but I shouldn't say what inspired me)

Calculus Level 4

\[\large \mathscr{E} = \displaystyle {\huge \int \limits_{0}^{\sqrt{3}}} \frac{2(1+x^2) - (2x)^2}{(1+x^2)^2 \sqrt{1- \left(\dfrac{2x}{1+x^2} \right)^2}} \, dx \]

If \( \mathscr{E} \) can be represented in the form \( \dfrac{a\pi}{b} \), with \( a\) and \(b \) being coprime positive integers, find \( a + b \).

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